Rows: 250
Columns: 16
$ Density <dbl> 1.0708, 1.0853, 1.0414, 1.0751, 1.0340, 1.0502, 1.0549, 1.0704…
$ Pct.BF <dbl> 12.3, 6.1, 25.3, 10.4, 28.7, 20.9, 19.2, 12.4, 4.1, 11.7, 7.1,…
$ Age <int> 23, 22, 22, 26, 24, 24, 26, 25, 25, 23, 26, 27, 32, 30, 35, 35…
$ Weight <dbl> 154.25, 173.25, 154.00, 184.75, 184.25, 210.25, 181.00, 176.00…
$ Height <dbl> 67.75, 72.25, 66.25, 72.25, 71.25, 74.75, 69.75, 72.50, 74.00,…
$ Neck <dbl> 36.2, 38.5, 34.0, 37.4, 34.4, 39.0, 36.4, 37.8, 38.1, 42.1, 38…
$ Chest <dbl> 93.1, 93.6, 95.8, 101.8, 97.3, 104.5, 105.1, 99.6, 100.9, 99.6…
$ Abdomen <dbl> 85.2, 83.0, 87.9, 86.4, 100.0, 94.4, 90.7, 88.5, 82.5, 88.6, 8…
$ Waist <dbl> 33.54331, 32.67717, 34.60630, 34.01575, 39.37008, 37.16535, 35…
$ Hip <dbl> 94.5, 98.7, 99.2, 101.2, 101.9, 107.8, 100.3, 97.1, 99.9, 104.…
$ Thigh <dbl> 59.0, 58.7, 59.6, 60.1, 63.2, 66.0, 58.4, 60.0, 62.9, 63.1, 59…
$ Knee <dbl> 37.3, 37.3, 38.9, 37.3, 42.2, 42.0, 38.3, 39.4, 38.3, 41.7, 39…
$ Ankle <dbl> 21.9, 23.4, 24.0, 22.8, 24.0, 25.6, 22.9, 23.2, 23.8, 25.0, 25…
$ Bicep <dbl> 32.0, 30.5, 28.8, 32.4, 32.2, 35.7, 31.9, 30.5, 35.9, 35.6, 32…
$ Forearm <dbl> 27.4, 28.9, 25.2, 29.4, 27.7, 30.6, 27.8, 29.0, 31.1, 30.0, 29…
$ Wrist <dbl> 17.1, 18.2, 16.6, 18.2, 17.7, 18.8, 17.7, 18.8, 18.2, 19.2, 18…Body Fat Percentage Presentation
Group 013E01
Joanne Lim, John Fu, Haoyuan Gao, Zhenchen Yi
Data Description
- Accurately body fat measurement is vital but often inconvenient and expensive
- Can we use simpler measurements like height, weight, and circumferences for estimates?
- The data set bodyfat contains body fat percentages and other related measurements for 250 men. These measurements, which include height, weight and various body circumferences, were collected to explore alternatives to underwater body fat assessments.
Check Linearity
Null and Full Model
Call:
lm(formula = Pct.BF ~ ., data = bodyfat)
Residuals:
Min 1Q Median 3Q Max
-8.3746 -0.3725 -0.1157 0.2358 15.0629
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.494e+02 1.154e+01 38.961 <2e-16 ***
Density -4.098e+02 8.384e+00 -48.876 <2e-16 ***
Age 1.395e-02 9.721e-03 1.435 0.153
Weight 1.527e-02 2.015e-02 0.758 0.449
Height -1.558e-02 5.752e-02 -0.271 0.787
Neck -1.653e-02 7.084e-02 -0.233 0.816
Chest 1.790e-02 3.259e-02 0.549 0.583
Abdomen 1.833e-02 3.286e-02 0.558 0.578
Waist NA NA NA NA
Hip 2.537e-02 4.391e-02 0.578 0.564
Thigh -2.107e-02 4.421e-02 -0.476 0.634
Knee -1.657e-02 7.366e-02 -0.225 0.822
Ankle -8.160e-02 6.616e-02 -1.233 0.219
Bicep -5.256e-02 5.132e-02 -1.024 0.307
Forearm 1.405e-02 6.229e-02 0.225 0.822
Wrist -1.883e-02 1.640e-01 -0.115 0.909
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.276 on 235 degrees of freedom
Multiple R-squared: 0.9777, Adjusted R-squared: 0.9763
F-statistic: 734.4 on 14 and 235 DF, p-value: < 2.2e-16Backward and Forward stepwise selection
Call:
lm(formula = Pct.BF ~ Density + Age + Abdomen, data = bodyfat)
Residuals:
Min 1Q Median 3Q Max
-8.2913 -0.3576 -0.0911 0.2319 15.4601
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.424e+02 8.738e+00 50.626 < 2e-16 ***
Density -4.065e+02 7.279e+00 -55.844 < 2e-16 ***
Age 1.182e-02 6.579e-03 1.796 0.0737 .
Abdomen 5.761e-02 1.332e-02 4.326 2.21e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.26 on 246 degrees of freedom
Multiple R-squared: 0.9772, Adjusted R-squared: 0.9769
F-statistic: 3513 on 3 and 246 DF, p-value: < 2.2e-16
Call:
lm(formula = Pct.BF ~ Density + Abdomen + Age, data = bodyfat)
Residuals:
Min 1Q Median 3Q Max
-8.2913 -0.3576 -0.0911 0.2319 15.4601
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.424e+02 8.738e+00 50.626 < 2e-16 ***
Density -4.065e+02 7.279e+00 -55.844 < 2e-16 ***
Abdomen 5.761e-02 1.332e-02 4.326 2.21e-05 ***
Age 1.182e-02 6.579e-03 1.796 0.0737 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.26 on 246 degrees of freedom
Multiple R-squared: 0.9772, Adjusted R-squared: 0.9769
F-statistic: 3513 on 3 and 246 DF, p-value: < 2.2e-16
Call:
lm(formula = log(Pct.BF) ~ Density + Age + Weight + Height +
Neck + Chest + Abdomen + Waist + Hip + Thigh + Knee + Ankle +
Bicep + Forearm + Wrist, data = bodyfat)
Residuals:
Min 1Q Median 3Q Max
-1.83722 -0.06872 0.03803 0.10425 1.10099
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 30.327682 1.893223 16.019 < 2e-16 ***
Density -28.194739 1.390322 -20.279 < 2e-16 ***
Age 0.003161 0.001595 1.982 0.04865 *
Weight -0.002840 0.003325 -0.854 0.39396
Height 0.028504 0.009489 3.004 0.00296 **
Neck -0.004012 0.011622 -0.345 0.73027
Chest 0.002934 0.005412 0.542 0.58830
Abdomen -0.002992 0.005408 -0.553 0.58062
Waist NA NA NA NA
Hip -0.007137 0.007208 -0.990 0.32311
Thigh 0.018438 0.007295 2.528 0.01215 *
Knee 0.001581 0.012085 0.131 0.89603
Ankle -0.011140 0.010860 -1.026 0.30605
Bicep 0.006681 0.008440 0.792 0.42940
Forearm -0.001098 0.010233 -0.107 0.91463
Wrist 0.021332 0.026949 0.792 0.42941
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2094 on 234 degrees of freedom
Multiple R-squared: 0.863, Adjusted R-squared: 0.8548
F-statistic: 105.3 on 14 and 234 DF, p-value: < 2.2e-16Summary of all 5 models
- We observed that the relationship between Neck, Abdomen, Bicep, Forearm, Ankle, and Pct.BF has a non-linear relationship, so we will transform these variables when we perform linear-log and log-log models.
| Full | Backward | Forward | Log-Linear | Linear-Log | Log-Log | |||||||
| Predictors | Estimates | p | Estimates | p | Estimates | p | Estimates | p | Estimates | p | Estimates | p |
| (Intercept) | 449.43 | <0.001 | 442.38 | <0.001 | 442.38 | <0.001 | 30.33 | <0.001 | 446.75 | <0.001 | -9.57 | 0.136 |
| Density | -409.76 | <0.001 | -406.49 | <0.001 | -406.49 | <0.001 | -28.19 | <0.001 | -413.59 | <0.001 | -27.40 | <0.001 |
| Age | 0.01 | 0.153 | 0.01 | 0.074 | 0.01 | 0.074 | 0.00 | 0.049 | 0.01 | 0.152 | 0.00 | 0.080 |
| Weight | 0.02 | 0.449 | -0.00 | 0.394 | 0.01 | 0.647 | 0.00 | 0.435 | ||||
| Height | -0.02 | 0.787 | 0.03 | 0.003 | -0.00 | 0.988 | 0.01 | 0.342 | ||||
| Neck | -0.02 | 0.816 | -0.00 | 0.730 | ||||||||
| Chest | 0.02 | 0.583 | 0.00 | 0.588 | 0.03 | 0.286 | 0.00 | 0.588 | ||||
| Abdomen | 0.02 | 0.578 | 0.06 | <0.001 | 0.06 | <0.001 | -0.00 | 0.581 | ||||
| Hip | 0.03 | 0.564 | -0.01 | 0.323 | 0.03 | 0.462 | -0.00 | 0.747 | ||||
| Thigh | -0.02 | 0.634 | 0.02 | 0.012 | -0.01 | 0.842 | 0.01 | 0.190 | ||||
| Knee | -0.02 | 0.822 | 0.00 | 0.896 | -0.02 | 0.831 | -0.01 | 0.522 | ||||
| Ankle | -0.08 | 0.219 | -0.01 | 0.306 | ||||||||
| Bicep | -0.05 | 0.307 | 0.01 | 0.429 | ||||||||
| Forearm | 0.01 | 0.822 | -0.00 | 0.915 | ||||||||
| Wrist | -0.02 | 0.909 | 0.02 | 0.429 | 0.02 | 0.888 | 0.02 | 0.441 | ||||
| Neck [log] | -0.88 | 0.737 | -0.60 | 0.135 | ||||||||
| Abdomen [log] | 4.00 | 0.738 | 12.41 | <0.001 | ||||||||
| Waist | -0.08 | 0.805 | -0.35 | <0.001 | ||||||||
| Ankle [log] | -2.12 | 0.207 | -0.42 | 0.101 | ||||||||
| Bicep [log] | -1.96 | 0.231 | 0.07 | 0.774 | ||||||||
| Forearm [log] | 0.72 | 0.670 | -0.14 | 0.599 | ||||||||
| Observations | 250 | 250 | 250 | 249 | 249 | 249 | ||||||
| R2 / R2 adjusted | 0.978 / 0.976 | 0.977 / 0.977 | 0.977 / 0.977 | 0.863 / 0.855 | 0.978 / 0.977 | 0.886 / 0.879 | ||||||
| AIC | 847.962 | 831.066 | 831.066 | 1353.555 | 836.172 | 1308.930 | ||||||
Model selection conclusion
- Generally, higher values of R-squared are better, but a very high R-squared could suggest overfitting, especially if it is much higher than the adjusted R-squared.
- Lower AIC values indicate a better-fitting model.
- We can conclude from the table, backward selection model will be the most appropriate one as it has the lowest AIC values and its R-squared and adjusted R-squared is the same.